Numerical solution of breakage population balance equations using differential algebraic equations form

N ageswara Rao N arni Department of Mathematics,

Rajiv Gandhi University of Knowledge Technologies, Gachibowli, Hyderabad, India 500032

Email: narninrao@gmail.com

Abstract-In this paper a new numerical approach is formulated to solve the breakage population balance equations along with its moments. In this approach the breakage equations are reformulated to a new class of differential equations known as differential algebraic equations. These equations involve differential equa­tions along with algebraic constraints. Here we consider the moment preserving constraint as the algebraic con­straint. The resulting differential algebraic equations are solved numerically using multistep methods of the MATLAB ODE suite.

I. INTRODUCTION Population Balance Equations (PBEs) are used in a

number of diverse fields such as granulation, crystalliza­tion, fluidized beds, atmospheric sciences, etc. Breakage is the process of splitting large particles into smaller ones. It occurs in a variety of processes such as polymer degradation, break of liquid droplets, aerosols, crushing of rocks, etc.

The time evolution of the breakage process depends qualitatively on the behaviour of the probability of break up for small particles and the selection of particles to break. To model this breakage process population balance equations are widely used. Analytical solutions for break­age population balance equations are available only for a few simple cases. Solving the breakage equation is still a challenging task. This paper uses a new form of breakage population balance equation to solve them.

The continuous binary breakage population balance equation [11] for a well mixed system is described by the integro partial differential equation

8n(t,x) 8t 100 (3(x, y)n(t, y)dy - S(x)n(t,x). (1)

The breakage kernel (3 (x, y) is assumed to be bounded for finite values of the volume coordinates x and y. The breakage kernel is often defined as the product of breakage function and selection function i.e., (3(x,y) = b(x,y) S(y). The breakage function b( x, y) is the probability density function for the formation of particles of size x from particles of size y, the selection function S describes the rate at which particles are selected to break.

The first term of the equation (1) is called as the birth term. From Mathematical point of view, this is a linear integral operator with variable limits and is classified as

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a linear Volterra integral operator. The second term is called as the death term. The breakage process (1) can be represented graphically as

+

Fig. 1. Breakage dynamics of population balance equation (1).

In the Figure 1, the particle of large size breaks into two particles of small size. As a result of it, two new particles of small size are born and a single particle of large size is disappeared from the system. Therefore in this breakage process, the particles flow towards the lower size range.

The breakage equation (1) is solved along with the initial and boundary conditions given as

n(O,x) n(t,O)

no(x), x E (0,00), 0, t 2: 0,

(2)

and the second condition indicates that there are no particles of zero size.

The breakage equation ( 1) can only be solved for analytically for very simple forms of breakage and selection functions, see [ 12]. Therefore we need to look for a nu­merical method to solve it. There exists various numerical method to solve it and are fall into stochastic methods [7], finite element methods [4], moment methods [5], and sectional methods [6].

In the population balance equation (1), the volume variable x range from ° to 00. In order to apply a numerical scheme for the solution of the equation a first step is to fix a finite computational domain. In this work we consider the following truncated equation

8n(t x) lxmax 8" = (3(x, y)n(t, y)dy - S(x)n(t, x), t x

with n(O,x) = no(x), x E 11:= (O,xmax).

(3)

As we can observe that the truncated breakage equation (3) has Volterra integral operator. The existence, unique­ness, and conservativeness of solutions for this truncated

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equation are discussed widely in the literature by various authors. For more details on this readers can refer [2], [3].

But the major drawback of this version is the conserva­tiveness. The solution obtained with the above formulation may not satisfy the moments of the population balance equations accurately. In order to overcome this difficulty we propose a new formulation of the population balance equations in such a way that the moments are satisfied and are obtained along with the population balance equations.

The paper is organized as follows: In Section II, the discretization of breakage population balance equations is discussed and a new set of equations are formed. In the Subsections II-A, II-B, the concepts of index calculations and numerical implementations for the newly formed dif­ferential algebraic equations are elaborated. Section III contains numerical examples and the simulation results with this new approach are analysed. At the end, conclu­sions are drawn and future challenges were discussed.

II. A NEW FORM OF BREAKAGE EQUATION A general one dimensional breakage population balance

equations for a well mixed system is given as

8n(t,x) roo

8t =

Jx b(x,y) S(y)n(t,y)dy - S(x)n(t,x), (4)

where n(t, x) is the number concentration, x, y are volume of particles, and t is the time. We can observe that the breakage function b(x, y) is symmetric b(x, y) = b(y, x) and satisfies the following properties:

lx

b(y,x)dy lI(x), b(x,y) = 0, fory>x

lx

yb(y,x)dy x.

The function lI(x) represents the number of fragments obtained from the breakage of the particle of size x. The second equality states that the total mass of the fragments equals the original mass when a particle of mass x breaks.

The invariant constraints of the breakage equation are 1. Zeroth moment, i.e., Total number of particles are ma

ma = 100 n(t, x)dx (5)

2. First moment, i.e. , Total volume is constant m 1

m1 = 100 xn(t,x)dx (6)

3. In general the rth moment is defined as

mr= 100 xrn(t,x)dx, r = 0, 1, 2, . . . (7)

The breakage population balance equations are ap­proximated by sectional methods like fixed pivot method, cell average technique, . . . ,etc. on a discrete uniform grid in space.(Volume, Length, . . . )

Consider the grid in ]E.1

!1h = {x E ]E.1 : x = hn, n E Zl }

where h is an arbitrary positive number (mesh step).

Let ni(t) = n(t,xi), where Xi E !1h, for i = 1, 2, 3, ... ,I, and xmax = Ih. The discrete version of the number based breakage equation is obtained as

dNi(t) 00 � = L. bij SjNj(t) - SiNi(t), for i = 1, 2, 3, ... ,I,

(8) where Ni(t) = N(t,Xi) = !::'Xin(t,Xi) are the number of particles per unit volume, subject to the invariant constraint

1. Zeroth moment, i.e. , Total number of particles are mo

I mo = LNi(t).

i=l Therefore the complete system of breakage equations along with its zeroth moment constraint is obtained as

dNi(t) 00 -- = '\"' b· S·N·(t) - SN·(t) dt � 'J J J " ,

I 0 = LNi(t) - mo·

i=l In the matrix form, we can write them as

1 o o

o o

o 0 1 0 o 1

o 0 o 0 o 0

o o

o o

1 0 o 0 ma L:�=1 bij SjNj(t) - SlN1(t)

L:f=2 bij SjNj(t) - S2N2(t) L:j=3 bij SjNj(t) - S3N3(t)

L: :=I bij SjNj(t) - SIN1(t) L:;=1 Ni(t) - mo

(9)

( 10)

Here' represents the derivative of N(t) with respect to the time t.

A. Index of the Breakage DAE

The index of the Differential Algebraic Equation mea­sures the singularity of the differential equation. There are different index concepts exists in the literature [8], [ 1], but two types of indexes are currently widely used differenti­ation index and perturbation inde.T. For the present DAEs differentiation index is used and is calculated as follows:

The system of breakage DAEs ( 10) formulated in Sec­tion II is of linear type. The matrix form of the breakage DAE is

M(t)N' = F(t,x,N), ( 11)

2013 International Conference on Advances in Computing, Communications and Informatics (ICACCl) 1385

where M(t) E 1R(I+1)x(J+l) is a singualr matrix known as mass matrix and is independent of t. Therefore the given system is a DAE rather than an ODE. The unknown vector is given as N = [N1(t),N2(t), ... ,N1(t),mo]T E lRJ+l, and the right hand side is obtained using any standard quadrature rule and is of the form F [!I(t),h(t), ... , /J(t), /J+1 V E lRJ+l.

The differentiation index is the minimum number of times that all or part of ( 11) must be differentiated with respect to t in order to determine 'LJ: as a continuous function oft and x. By differentiating of (I + l)th equation with respect to t, we get the differential equation

dNI +

dN2 +

. . . + dN] _ d mo = 0.

dt dt dt dt Therefore from the above calculations it is clear that the Breakage DAE is of differentiation index 1. So we can solve this system by using any numerical method that suite for index 1 DAEs. For the present work we consider the methods based on Backward Differentiation Formulas (BDF). In the next Subsection II-B, the details of the numerical methods and their numerical implementations are given.

B. Numerical methods

A part of MAT LAB ODE suite known as ode15s is used to solve the DAEs. This is based on the Gear's method for solving the linear DAEs of stiff type. The ode15s code is a quasi-constant step size implementation in terms of back­ward differences of the Klopfenstein-Shampine family of Numerical Differentiation Formulas (NDF's). More details of this suite can be found in [9], [10].

III. NUMERICAL EXAMPLES Example 1 Let us consider a breakage equation with

uniform binary breakage function b( x, y) = t and a linear selection function S(x) = x. The breakage equation (1) with these terms is written as

8n(t,x) 1002 8

= - xn(t,y)dy-xn(t,x), t x y

subject to the initial and boundary conditions

( 12)

n(0,x)=5(x-L), n(t,O) =0. (13)

The analytical solution of the above equation ( 12) along with its initial and boundary is conditions (13) is obtained by Ziff and McGrady in [ 12] and is given as

n(t, x) = exp( -tx){ 5(x -L) + [2t + t2(L - x)] 8 (L -x)} , (14)

where 8 is the step function defined as

8 ( -L)- { 1, x<L, x - 0, otherwise,

and 5(x -L) is the Dirac delta function defined as

£ ( L) { I, if x = L, u x - = 0, otherwise.

Let us consider the discretization of the domain with nonuniform mesh i.e., with geometric mesh defined as

Xi = d ixmin, for i = 1, 2, 3, ... , I, (15) where xmin is the minimum particle size considered and define the xmax as the maximum particle size of the spectrum, which is equal to d I - 1 x xmin . The geometric factor d is calculated as

1

d = (xm�x ) I-1

. xmzn The grid points are numbered from small to large sizes with Xi < Xi+1 , i = 1, 2, 3, ... , I as shown in the Figure 2.

xl�I --�x·· 2�---X�3·'-------X��'--------x-r·��I---------- X-iJ Fig. 2. Grid system for the discretized equations.

The discrete version of the number based truncated breakage equation is obtained as

dNi(t) dt

Ni(O)

Fi(= Ii -XiNi(t)), (16)

5(Xi-L), fori = I, 2, 3, ... I, N1(t) 0, t 2': 0,

where Ii is the value of Volterra integral obtained after discritizing the integral with trapezoidal rule. The ze­roth moment mo is added as a new additional variable to the existing equation ( 16) . It was denoted as NJ+l. The consistent initial condition for this new variable is obtained from the given problem. For the given problem, the consistent initial condition is calculated as NJ+l = 1. Therefore the Differential Algebraic Equations form of Breakage population balance equation is formulated as { dNi(t) � = Fi(= Ii - .TiNi(t)),

° = LNi -NJ+l i=1

for i = 1, 2, 3, ... I.

(17)

and the consistent initial and boundary conditions are {Ni(0)=5(Xi-L), fori = I, 2, 3, ... I, NJ+l(O) = 1, N1(t) = 0, for t 2': 0,

(18)

Thus a Differential Algebraic Equation form (17), (18) of Breakage equation (16) is formulated. It was calculated that the differentiation index of this Breakage Differential Algebraic Equation system is one. Therefore, an ODE suite of MATLAB, ode15s is used to solve the DAE system.

The system (17) of breakage DAEs are solved from t = ° sec to t = 60 sec on a geometric grid of size 30. The

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volume domain is extended from Xl = 10-9 to X30 = 1. It should be noted that the (1 + l)th initial and boundary conditions are supplied in order to get a consistent system of DAEs. The number density is plotted in the Figure 3 along with its analytic solution (14).

4000 ,-----,--------r-----,-----;======il

3500

3000

>, 2500 .., '[iJ 5 2000

"D iil 1500

.0

§ 1000

Z 500

-500'-:-_'o::-----'-:c-------L::------'--:-------'--:------' 10 10-8 10-6 10-4 10-' 10°

Volume x

Fig. 3. Particle size distribution on semi-log scale for mono-disperse initial condition.

It can be observed from Figure 3 that the Numerical solution approximated the analytical solution accurately. The CPU time for this computation is 4.14596 Seconds. From this new approach we can find the zeroth moment along with the particle size distribution. For this problem the zeroth moment along with the solution is calculated and is labelled as an extra variable mo = NIH' The zeroth moment obtained from the simulation and from the analytical solution are plotted in the Figure 4.

6000 ,---,-----.--,-----,----,--,----.,======il.

5000 o .: 4000 .: <I)

§ 3000 ::8 ..c: C 2000 '"' <I)

N 1000

3 4 5 6 Timet

7 9 10

Fig. 4. Temporal change of zeroth moment for mono-disperse initial condition.

It is observed that the zeroth moment obtained from this new formulation deviates from the analytical solution as t increases. But the deviation is significantly less compared to other methods [6].

ExaIllple 2 Let us now consider a second example for breakage equation with uniform binary breakage function b( x, y) = t and a linear selection function S (x) = x. The breakage equation (1) with these terms is reduced to

8n(t,x) 100 2 8

= -xn(t,y)dy- xn(t,x), t x y

(19)

subject to the exponential initial and boundary conditions

n(O,x) = exp(-x), n(t, 0) = 0. (20)

The analytical solution of the above equation (19) along with its exponential initial and boundary is conditions (20) is obtained by Ziff and McGrady in [12] and is given as

n(t,x) =exp(-x(l +t))(I +t)2. (21)

The Breakage population balance equation is discretized according to the previous calculations (17) and (18). The differentiation index of the resulting DAEs is one. For this new formulation initial and boundary conditions are applied as given in the equation (20). But in order to have a consistent initial condition for the (1 + l)th variable, the initial condition is calculated as

I I

N(O,I + 1) = LN(O,i) = Le-Xi.

i=l i=l

The resulting discretized system is solved on a geomet­ric grid of size 60 with a minimum volume xmin = 10-9 and maximum volume xmax = 125. The results are plotted in the Figure 5 after t = 10 seconds of simulation along with its analytical solution (21).

140

120 1- Analytical "'Numerical

.: 100 >, '@ 80 <I)

"0 60 '"' <I)

.0 40 §

Z 20

-20 '-:-, o::----'-:c------'--;:-----'--:-----'--;;-----'-:c---'-;;-----' 10- 10-8 10-6 10-4 W' 10° 10' 104

Volume x

Fig. 5. Particle size distribution on semi-log scale for exponential initial condition.

From the Figure 5 it is observed that the number densi­ties calculated from the numerical method are slightly over predicted than the analytical ones at small size particles. The CPU time for this computation is 2.48357 seconds. The value of the zeroth moment is obtained along with the numerical solution as an additional variable mo = NIH

and is plotted in the Figure 6 along with its analytical calculations.

2013 International Conference on Advances in Computing, Communications and Informatics (ICACCl) 1387

gX10

7 0

� 6 � S

5 0

:;E4 ..c:

'0 3 '""' (l)

N 2

5

30 40 50 60 Timet

70 80 gO 100

Fig. 6. Temporal change of zeroth moment for exponential initial condition.

The above Figure 6 shows that the zeroth moments are calculated accurately for small t. As t increases the zeroth moment value deviated significantly from the analytical calculation. This is due to the complexity in the initial condition.

IV. CONCLUSIONS

In this paper, a new approach to solve the breakage equation ( 1) is proposed. In this approach, a Differential Algebraic Equations system is solved satisfying the zeroth moment constraint. The differentiation index of this new DAE system is calculated and is equal to one. The new for­mulation has the advantage of finding the zeroth moment along with the solution. It is interesting to note that the CPU time for this new formulation is very less and still a complete analysis of the performance of this formulation is a future task.

The present method can be extended further with the inclusion of the higher order moments. Inclusion of higher order moments has the advantage of better prediction of the particle size distribution. This may result to form a high index DAEs. But from computation point of view, it may be difficult to find a suitable numerical method to solve the resulting high index DAE system. At present work is under progress to include the first moment i.e., the conservation of volume.

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